Notes: Linear Probability Model

linear probability model $Y_i=\beta_1+\beta_2 X_i+\mu_i$where Y takes two values:0 and 1. The probabilities are
$Pr(Y_i=1|X_i)=\beta_1+\beta_x X_i$

• the mean of error term $\mu_i$
  Since y is binary variables which takes two values 0 and 1 with probability 1,
  $E(\mu_i)=p_1\mu_1+p_0\mu_0\\ =[\beta_1 +\beta_2 X_i][1-(\beta_1 +\beta_2 X_i)]+[1-(\beta_1 +\beta_2 X_i)][-(\beta_1 +\beta_2 X_i)]\\=0$

• the variance of error term $\mu_i$
  since E( $\mu_i$)=0,
  $Var(\mu_i)=p_0\mu_0^2+p_1\mu_1^2\\ =[1-(\beta_1 +\beta_2 x)][-(\beta_1 +\beta_2 x)]^2+[\beta_1 +\beta_2 x][1-(\beta_1 +\beta_2 x)]^2\\ =[\beta_1 +\beta_2 x][1-(\beta_1 +\beta_2 x)]$

• the linear probability model is not the best model and therefore a Probit or a Logit model is used.
  in LPM, there is no guarantee that the probability would be in the range of [0,1]; and the marginal effect in LPM is constant which is not practical.

Microeconometrics course, University of LIVERPOOL